# Two correlated marginally Gaussian RV, but not Jointly Gaussian

We know if two RVs are marginally Gaussian and uncorrelated, they can be not jointly Gaussian. How about two correlated RVs which are marginally Gaussian? Are they Jointly Gaussian or no, there are occasions when they are not Jointly Gaussian?

• Are you asking for an example where $X$ is normal, $Y$ is normal, $\mathrm{Cov}(X,Y) \neq 0$ and $(X,Y)$ not jointly normal? – student Sep 29 '17 at 19:18

Suppose $U,V$ are i.i.d. $N(0,1).$ Let $(X,Y)=(U,U)$ if $U>0$ and let $(X,Y)=(-|U|,-|V|)$ if $U\le0.$
Or let $Z\sim N(0,1)$ and let $(X,Y)=(Z,Z)$ if $|Z|>1$ and let $(X,Y)=(Z,2-Z)$ if $|Z|\le 1.$
In both cases, $X$ and $Y$ are marginally gaussian, correlated, but not jointly gaussian.